${\displaystyle 2\gamma _{a}\sigma T_{a}^{4}=\gamma _{s}\sigma T_{s}^{4}}$ (4j)
 ${\displaystyle {\sqrt[{4}]{\tfrac {S_{o}(1-\sigma )}{4\sigma (1-{\frac {\gamma _{a}}{2}})}}}=T_{s}}$ (4k)
With no atmosphere at all, ${\displaystyle \gamma _{a}=0}$ and the equation above just reduces to our original equation, giving us an answer of 255K. By plugging in the observed temperature at the Earth's surface (288K) and solving for ${\displaystyle \gamma _{a}}$, we obtain a value of ${\displaystyle \gamma =.76}$. With that value in hand, then, we can actually use this model to explore the response of the planet to changes in albedo or greenhouse gas composition—we can make genuine predictions about what will happen to the planet if our atmosphere becomes more opaque to infrared radiation, more energy comes in from the sun, or the reflective profile of the surface changes. This is a fully-developed ZDEBM, and while it is only modestly powerful, it is a working model that could be employed to make accurate, interesting predictions. It is a real pattern.